1. Field of the Invention
The present invention relates to beamforming by receiving signals with sensors in an array, and more particularly to beamforming with such arrays when wideband signals are received.
2. Description of the Prior Art
Sonar, radar, communications, seismological prospecting, and tomography systems employ arrays of spatially distributed sensors which sample physical quantities such as pressure or electromagnetic fields and convert these quantities to electrical signals. These electrical signals are processed to produce a second set of signals that enhance wave arrivals from selected directions, while discriminating against wave arrivals from other directions, thereby forming beams in the selected directions. This process is known in the art as beamforming and the network to which the sensors are coupled is known as a beamforming network. Signals from the sensor coupled to the beamforming network may be continuous or sampled analog waveforms or they may be sampled and digitized to establish digital signals. Beamforming with analog signals requires that each signal be time delayed in accordance with the desired beam direction and the position of the receiving sensor in the array, amplitude weighted in accordance with the beam shape desired, and added with the other signals to form a beam output signal. These time delays, weightings, and sum operations are generally duplicated for each selected beam direction. When the signals received by the sensors are at frequencies within a narrow band centered about a carrier frequency f.sub.0, the required delay operations may be performed by lumped constant phase-shift circuits that provide phase shifts in accordance with .phi.=2.pi.f.sub.0 .tau..sub.k, where .tau..sub.k is a function of the selected direction and k is an integer index corresponding to the receiving sensor.
Sensor output signals may be directly sampled or may first be hetrodyned to a convenient intermediate frequency and then sampled. Alternatively, a pair of signals may be derived which represent the in-phase and quadrature signal components relative to the carrier frequency f.sub.0, each such signal being sampled. The signal sample pairs thus produced may be considered a complex-valued signal sample S.sub.k (n.DELTA.t), derived from the kth array element, where n is the time sample index and .DELTA.t the sample period. For receptions which are narrowband about the carrier frequency f.sub.0, these time-sampled signals are phase-shifted and summed to form a beam in accordance with ##EQU1## where B.sub.m (n.DELTA.t) is the mth beam output signal, S.sub.k (n.DELTA.t) is the sampled signal from the kth sensor, a.sub.k is the weighting or shading factor for the signal from the kth sensor, and .phi..sub.km the phase-shift value required to phase-align the signal from the kth sensor with the signals from all the other sensors for the mth beam selection direction. It is well known that the sampling rate (.DELTA.t).sup.-1 must exceed the bandwidth W of the sensor output signal about the carrier frequency.
Signal samples produced by a uniform plane wave at the carrier frequency, arriving at an angle .theta., at the kth element of an array of K sensors linearly positioned with uniform spacing of d wavelengths at the center frequency f.sub.0 may be represented as S.sub.k (n.DELTA.t)=Ae.sup.-j2.pi.kd sin .theta. where A is the wave amplitude. If the sensor signals are subjected to phase shifts .phi..sub.km =(2.pi.km)/K applied thereto and then summed, m being a constant that may take on the values 0, 1, 2, . . . , (K-1), the array will be steered to couple signals from the sensors for summations that are of equal phase for plane wave fronts at the carrier frequency arriving at angles defined by .theta..sub.m =sin.sup.-1 (m/Kd). With this phase gradient the sum of the sample signals B.sub.m (n.DELTA.t) becomes ##EQU2## which is well known in the art as the discrete Fourier tranform (DFT). When the frequency band of the signals S.sub.k received at the sensors is sufficiently broad about the carrier frequency, beam steering as described above fails to operate properly since the phase shift values at the elements, though based on the propagation delays of the wave front as it crosses the array, do not provide proper phase shifts for signal components at frequencies sufficiently far removed from the carrier.
Consider steering a uniform colinear array to a direction .theta..sub.m =sin.sup.-1 (m/Kd) for a wave at the carrier frequency f.sub.0. The phase shifts required for the kth sensor in the beamforming process are thus ##EQU3## where .lambda.=c/f.sub.0 is the wavelength at the carrier frequency and c is the wave propagation speed. When the wave arriving from .theta..sub.m has a temporal frequency f.sub.0 +.DELTA.f, it induces a relative phase shift at the kth sensor of ##EQU4## and the phase shifter at each element no longer exactly compensates for the propagation-induced phase shift. In fact a beam for a selected angle .theta..sub.m under the assumption of the frequency f.sub.0, is steered to the angle ##EQU5## for an incident wave at frequency f.sub.0 +.DELTA.f. This defocusing effect causes the response of the phase-shift beamformer to encompass a broader spatial angle, provides a diminished beam amplitude, and causes adjacent beams to smear together, resulting in a loss of directional resolution. Thus the maximum scan angle of a phase-shift steered array is a function of the array size and the operating signal bandwidth W, the maximum scan angle being given approximately by ##EQU6## where T.sub.a is the time required for a wave traveling parallel to the array elements to traverse the array and T.sub.a W is a fill-time/bandwidth product for the array.
The fill-time/bandwidth product scan angle limitation has been overcome in the prior art with sampled data versions of delay and sum beamforming. In one method sensor signals are sampled at a rate much faster than that required by the signal bandwidth, and beams are formed by selecting sensor samples corresponding to the required sensor delays for the desired beam angle of arrival. Another method utilized in the prior art, as described by R. G. Pridham and R. A. Mucci, "A Novel Approach to Digital Beamforming", Journal of the Acoustical Society of America, volume 63, pp. 425-434, February 1978, performs sampling at a rate that is slower than the above mentioned sampling rate to form estimates of the sensor signal samples at the desired delays via interpolation. Though these beamforming methods exhibit satisfactory performance with wideband signals, they are considerably more complex and expensive to build than phase-shift beamformers.